I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie.

In math line (2.1.12), Illusie considers the pairing $\langle-,-\rangle:\Omega_{X/S}^1\times T_{X/S}\longrightarrow \mathcal{O}_X$ of the tangent and cotangent bundles on a scheme $X$ of characteristic $p$, relative to a morphism $X\longrightarrow S$. ($S$ also has characteristic $p$).

Earlier we defined a bunch of maps, which would be necessary to describe in order for me to ask my question. The first one is the Cartier operation, $C$, which sends a closed form $w\in \Omega_{X/S}^1$ to a 1-form $Cw \in \Omega_{X^{(p)}/S}^1$, where $X^{(p)}$ is the base change of $X\longrightarrow S$ with respect to the absolute Frobenius on $S$. Denote by $W$ the canonical projection $W:X^{(p)}\longrightarrow X$.

Illusie then wants to say something about the pairing $$\langle Cw,W^*D\rangle$$ which happens on $\Omega_{X^{(p)}/S}^1\times T_{X^{(p)}/S}\longrightarrow \mathcal{O}_{X^{(p)}}$. Namely, he gives the identity $$\langle Cw,W^*D\rangle^p = \langle w,D^p\rangle - D^{p-1}\langle w,D\rangle.$$ My questions are:

What does the $p$-th power of a tangent vector, $D^p$, mean? Does the tangent space have a ring structure?

The pairing apriori should have values in $\mathcal{O}_{X^{(p)}}$, once it is raised to the $p$'th power we regard it as having values in $\mathcal{O}_X$ as this is. For this reason it seems that Illusie is regarding $D^{p-1}$ as an element of $\mathcal{O}_X$, which is consistent with math line (2.1.13), why is that? I guess this goes back to the first question.

Illusie also writes $D_i^p = 0$, where $D_i = \partial/\partial x_i$, for some etale basis $(x_i)$, why is that true? I guess that this should all be clear once I understand this notation.

Thanks in advance!